350 
-4e- 
TABLE II. 
Energy balance for T.N.T./Aluminum expressed in calories per gram of explosive. 
Position of Shock— 
Front. 
(Charge Radii). 
1.00 1.04 1.21 
ee Po 
ee ee 
The discrepancies in the energy in the latter stages of the work are due to a systematic 
cause explained in Section 6. It must not be assumed that the work has suddenly become less 
accurate, 
Phe method of calculating these energies is fairly obvious. The potential energies of 
gas and water were computed as functions of pressure from the equation of state (by means of the 
familiar integral p ba which gives the potential energy per unit volume). The kinetic energy 
PoP 
per unit volume is given simply by 3P ut , and u being know at every point, so that one can obtain 
the total kinetic energies by integrating this over the whole volume of the gas and the disturbed 
volume of the water, the weighting factor being 477 r-dr. Similar integrations give the two 
potential energmes. The total wastage was calculated from the figures given in Penney and Dasgupta’s 
paper Table |, relating @, the rate of wastage, to shock-wave pressure. The relations between time 
and position of shock-wave and value of shock-wave pressure being known, the wastage while the shock— 
h 
wave moves from R, to Ry is given by a r? 6 pdr. 
No 
Kirkwood(3), by a process that is not explicitly stated, concluded that his results were not 
likely to be ‘n error by more than 20%, and that they would be too high. This conclusion seems to be 
borne out by the comparison between theory and experiment, as will be seen later. Penney has 
questionec the validity of some of Kirkwood's assumptions and Kirkwood has replied answering some of 
the points raised. The only serious difficulty still outstanding is to decide to what extent one 
can replace, as Kirkwood does, the state of affairs represented by equations (1) and (2) by his rather 
simpler scheme of things, in which the outgoing wave in the gas and the ingoing wave in the water are 
both neglected. Calculations in fact show that the ingoing wave in the water is very far from 
negligible. it occurs for two reasons. First, because of tne term — 2 uc in equation (2), which 
Causes the Q function to build up negatively as the wave travels inwards. Secondly, because the 
shock-front acts in some ways as a reflector owing to the overtaking effect, so that Q is not 
precisely zero even just behind the shock-front. In Kirkwood's first paper (0.S.R.0,588) it was 
erroneously stated that the theory implies neglect of the Q function. This is not so, because 
Kirkwood’s function § is not precisely the same as the Riemann function P, and Kirkwood has pointed 
out that his results in fact imply values of Q of the same order of magnitude as those given by the 
step-by-step calculations. However, the procedure of replacing two waves proceeding in opposite 
directions by a single wave cannot be carried out even in the simple case of a stretched string, so 
it is hard tc see how it cantbe valid here. The comparative success of Kirkwaod's theory is 
probably due to two main causes:— 
(a) It is definitely a better approximation than “acoustic” theory. 
(b) Substitution in the hydrodynamic equations shavs that the first-order correction terms 
vanish, at all events for water. 
A secee 
